Fungrim home page

# Fungrim entry: 47cf5d

$R_G\!\left(x, y, z\right) = 2 R_G\!\left(x + \lambda, y + \lambda, z + \lambda\right) - \frac{1}{2} \left(\lambda R_F\!\left(x, y, z\right) + \sqrt{x} + \sqrt{y} + \sqrt{z}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)$
TeX:
R_G\!\left(x, y, z\right) = 2 R_G\!\left(x + \lambda, y + \lambda, z + \lambda\right) - \frac{1}{2} \left(\lambda R_F\!\left(x, y, z\right) + \sqrt{x} + \sqrt{y} + \sqrt{z}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
CarlsonRF$R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("47cf5d"),
Formula(Equal(CarlsonRG(x, y, z), Where(Sub(Mul(2, CarlsonRG(Add(x, lamda), Add(y, lamda), Add(z, lamda))), Mul(Div(1, 2), Add(Add(Add(Mul(lamda, CarlsonRF(x, y, z)), Sqrt(x)), Sqrt(y)), Sqrt(z)))), Def(lamda, Add(Add(Mul(Sqrt(x), Sqrt(y)), Mul(Sqrt(y), Sqrt(z))), Mul(Sqrt(x), Sqrt(z))))))),
Variables(x, y, z),
Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC